Stability analysis of switched multiple nonlinear discrete systems with interval time varying delays

STABILITY ANALYSIS OF SWITCHED MULTIPLE NONLINEAR<br /> DISCRETE SYSTEMS WITH INTERVAL TIME-VARYING DELAYS<br /> <br /> Tran Nguyen Binh∗ , Nguyen Thi Thu Huong, Pham Thi Linh<br /> Thai Nguyen University of Economics and Business Administration - Thai Nguyen University<br /> <br /> ABSTRACT<br /> <br /> This paper deals with the problem of asymptotic stability for a class of switched nonlineardiscrete systems<br /> with time-varying delay. The time-varying delay is assumed to be belong to a given interval in which the<br /> lower bound of delay is not restricted to zero. A linear matrix inequality (LMI) approach to asymptotic<br /> stability of the system is presented. Based on the Lyapunov functional, delay-depenent criteria for the<br /> asymptotic stability of the system are established via linear matrix inequalities.<br /> <br /> Keywords:<br /> <br /> Switching, discrete systems, uncertainty, Lyapunov function, linear<br /> <br /> matrix inequality.<br /> <br /> 1<br /> <br /> INTRODUCTION<br /> <br /> A switched system is a particular kind of hybrid<br /> system that consists of several subsystems and<br /> a switching law determining at any time instant<br /> which subsystem is active. There are indeed<br /> many switched systems that occur naturally or<br /> by design in the fields of control, communication, computer and signal processes. A different switching rule would cause different behavior of the system and hence lead to different system performances. Because of the complexity<br /> of the designing switched law for the systems,<br /> the stability analysis and control synthesis of<br /> switched systems becomes more difficulty and<br /> attracts the interest of several scientists during<br /> the last decades [111].<br /> On the other hand, time-delay phenomena are<br /> very common in practical systems. A switched<br /> system with time-delay individual subsystems<br /> is called a switched time-delay system; in particular, when the subsystems are linear, it is<br /> then called a switched time-delay linear system.<br /> During the past decades, the stability analysis of switched linear continuous/discrete timedelay systems has attracted a lot of attention<br /> [48]. The main approach for stability analysis<br /> relies on the use of LyapunovKrasovskii functionals and linear matrix inequality (LMI) ap0 *Tel:<br /> <br /> proach for constructing a common Lyapunov<br /> function [911]. Although many important results have been obtained for switched linear<br /> continuous-time systems, there are few results<br /> concerning the stability of switched linear discrete systems with time-varying delays. It<br /> was shown in [6,8,12] that when all subsystems are asymptotically stable, the switching<br /> system is asymptotically stable under an arbitrary switching rule. The asymptotic stability<br /> for switching linear discrete time-delay systems<br /> has been studied in [13], but the result was limited to constant delays.<br /> Compared to the existing results, our result has<br /> its own advantages. First, the time delay is assumed to be a time-varying function belonging<br /> to a given interval, which means that the lower<br /> and upper bounds for the time-varying delay<br /> are available, the delay function is bounded but<br /> not restricted to zero. Second, the approach<br /> allows us to design the switching rule for stability and stabilization in terms of LMIs, which<br /> can be solvable by utilizing Matlabs LMI Control Toolbox available in the literature to date.<br /> The paper is organized as follows: Section 2<br /> presents definitions and some well-known technical propositions needed for the proof of the<br /> main results. Switching rule for the asymptotic stability and stabilization is presented in<br /> Section 3.<br /> <br /> 0984411299, e-mail: nguyenbinh.tueba@gmail.com<br /> <br /> Notations.<br /> The following notations will be used throughout this paper. R+ denotes the set of all<br /> real non-negative numbers; Rn denotes the ndimensional space with the scalar product of<br /> two vectors hx, yi or xT y; Rn×r denotes the<br /> space of all matrices of (n × r)− dimension.<br /> AT denotes the transpose of A; a matrix A<br /> is symmetric if A = AT . Matrix A is semipositive definite (A ≥ 0) if hAx, xi ≥ 0, for<br /> all x ∈ Rn ; A is positive definite (A > 0) if<br /> hAx, xi > 0 for all x 6= 0; A ≥ B means<br /> A − B ≥ 0. λ(A) denotes the set of all eigenvalues of A; λmin (A) = min{Reλ : λ ∈ λ(A)};<br /> λmax (A) = max{Reλ : λ ∈ λ(A)}.<br /> <br /> . The time delay function h(k) satisfies the following condition 0 < h ≤ h(k) ≤ h, ∀k ∈ N+ ,<br /> where h, h are positive integers.<br /> Definition 2.1. The switched system (2.1) is<br /> asymptotically stable if there exists a switching<br /> function σ(.) such that the zero solution of the<br /> system is asymptotically stable.<br /> Definition 2.2.<br /> The system of matrices<br /> {Li }, i = 1, 2, ..., N, is said to be strictly completed if for every x ∈ Rn \{0} there is i ∈<br /> {1, 2, ..., N } such that xT Li x < 0.<br /> It is easy to see that the systems {Li } is strictly<br /> complete if and only if<br /> N<br /> [<br /> <br /> 2<br /> <br /> PRELIMINARIES<br /> <br /> Ωi = Rn \{0}<br /> <br /> i=1<br /> <br /> Consider an uncertain nonlinear discrete-time<br /> systems with time-varying delay of the form<br /> x(k + 1) =Aσ x(k) + Bσ x(k − hj (k))<br /> <br /> where<br /> Ωi = {x ∈ Rn : xT Li x < 0}, i = 1, 2, .., N.<br /> <br /> + fσ (k, x(k), x(k − h(k)), k ∈ Z+<br /> x(k) = φ(k),<br /> <br /> k ∈ [−h, −h + 1, ..., 0],<br /> (2.1)<br /> <br /> where x(k) ∈ Rn is the state; σ(.) : Rn −→<br /> {1, 2, ..., N } is the switching rule, which is a<br /> function depending on the state at each time<br /> and will be designed. A switching function is a<br /> rule which determines a switching sequence for<br /> a given switching system. Moreover, σ(x) = i<br /> implies that the system realization is chosen<br /> as the ith system, i = 1, 2, ..., N. It is seen<br /> that the system (2.1) can be viewed as an autonomous switched system in which the effective subsystem changes when the state x(k) hits<br /> predefined boundaries. Ai , Bi , i = 1, 2, ..., N<br /> are given matrices; φ(k) is the initial function<br /> with the norm<br /> <br /> The following technical lemmas will be used in<br /> the proof of the results.<br /> Lemma 2.1. [3] Let E, H and F be any constant matrices of appropriate dimensions and<br /> F T F ≤ I. For any  > 0, we have<br /> EF H + H T F T E T ≤ EE T + −1 H T H.<br /> <br /> Lemma 2.2. [5] The system {Li } is strictly<br /> N<br /> P<br /> complete if there exist ξi ≥ 0,<br /> ξi > 0 such<br /> i=1<br /> <br /> that<br /> <br /> N<br /> P<br /> <br /> ξi Li < 0, i = 1, 2, ..., N.<br /> <br /> i=1<br /> <br /> k φ k=<br /> <br /> max<br /> <br /> k φ(i) k;<br /> <br /> i∈[−h,−h+1,...,0]<br /> <br /> The nonlinear purterbation fi (k, x, x1 ) : Z+ ×<br /> Rn × Rn × 7−→ R+ satisfies the following condition:<br /> ∃G, H :fiT (k, x, x1 )fi (k, x, x1 ) ≤<br /> xT GT Gx + xT1 H T Hx1 , i = 1, 2, ..., N.<br /> (2.2)<br /> <br /> Lemma 2.3. [4] For any given vectors vi ∈<br /> RT , i = 1, 2, .., n, the following inequality holds:<br /> <br /> (<br /> <br /> n<br /> X<br /> i=1<br /> <br /> vi )T (<br /> <br /> n<br /> X<br /> i=1<br /> <br /> vi ) ≤ n<br /> <br /> n<br /> X<br /> i=1<br /> <br /> viT vi .<br /> <br /> 3<br /> <br /> MAIN RESULTS<br /> <br /> ∃ξi ≥ 0,<br /> <br /> N<br /> X<br /> <br /> ξi > 0 :<br /> <br /> i=1<br /> <br /> Let us set<br /> <br /> N<br /> X<br /> <br /> ξi Li (R, Ω, H, W, T, S) < 0,<br /> <br /> i=1<br /> <br /> i = 1, 2, ..., N.<br /> (3.3)<br /> <br /> i<br /> <br /> T =<br />  Ti<br /> <br /> <br /> <br /> <br /> <br /> <br /> <br /> <br /> <br /> <br /> i<br /> i1<br /> 11 T12 T13<br /> i<br /> i1<br /> ∗ T22<br /> T23<br /> i1<br /> ∗<br /> ∗ T33<br /> ∗<br /> ∗<br /> ∗<br /> ∗<br /> ∗<br /> ∗<br /> ∗<br /> ∗<br /> ∗<br /> <br /> i<br /> T14<br /> −P1<br /> i<br /> T34<br /> i<br /> T44<br /> ∗<br /> ∗<br /> <br /> P1T<br /> P1T<br /> P1T<br /> P1T −ΩT +P1T<br /> 0<br /> 0<br /> 0<br /> 0<br /> −ΩT<br /> −P1T −P1T −P1T −P1T<br /> −P1T<br /> −P1T −P1T −P1T −P1T<br /> −P1T<br /> −Q<br /> 0<br /> 0<br /> 0<br /> 0<br /> T<br /> ∗ − 1+h<br /> 0<br /> 0<br /> 0<br /> <br /> ∗<br /> <br /> ∗<br /> <br /> ∗<br /> <br /> ∗<br /> <br /> ∗<br /> <br /> ∗<br /> <br /> S<br /> − 1+h<br /> <br /> 0<br /> <br /> 0<br /> <br /> ∗<br /> ∗<br /> <br /> ∗<br /> ∗<br /> <br /> ∗<br /> ∗<br /> <br /> ∗<br /> ∗<br /> <br /> ∗<br /> ∗<br /> <br /> ∗<br /> ∗<br /> <br /> ∗<br /> ∗<br /> <br /> −R<br /> ∗<br /> <br /> −P1T<br /> −λI<br /> <br /> <br /> <br /> The switching rule is chosen as σ(x(k)) = i,<br /> whenever x(k) ∈ Ωi , i = 1, 2, ..., N.<br /> <br /> <br /> <br /> <br /> <br /> ,<br /> <br /> <br /> <br /> <br /> <br /> Proof. Consider the<br /> Krasovskii function<br /> <br /> Li (R, Ω, H, W, T, S) = −ΩT Ai − ATi Ω + R + H<br /> + (h − h)W + (1 + h)T + (1 + h)S + λGTi Gi ;<br /> Ωi = {x ∈ Rn : xT Li (R, Ω, H, W, T, S)x < 0},<br /> <br /> following<br /> <br /> V (x(k)) =V1 (x(k)) + V2 (x(k)) + V3 (x(k))<br /> + V4 (X(k)) + V5 (x(k)),<br /> where<br /> V1 (x(k)) = xT (k)P x(k);<br /> k−1<br /> X<br /> <br /> V2 (x(k)) =<br /> <br /> i = 1, 2, ..., N ;<br /> Ω1 = Ω1 , Ωi = Ωi \<br /> <br /> V3 (x(k)) =<br /> <br /> Ωj , i = 2, 3, ..., N,<br /> <br /> k−1<br /> X<br /> <br /> xT (i)Qx(i) +<br /> <br /> p<br /> X<br /> <br /> xT (i)Rx(i)<br /> <br /> i=k−h<br /> <br /> i=k−h<br /> i−1<br /> [<br /> <br /> Lyapunov-<br /> <br /> k−1<br /> X<br /> <br /> xT (i)Hx(i);<br /> <br /> j=1 i= k−hj (k)<br /> <br /> j=1<br /> <br /> −h+1<br /> <br /> where<br /> <br /> X<br /> <br /> V4 (x(k)) =<br /> <br /> λ > 0,<br /> <br /> k−1<br /> X<br /> <br /> xT (i)W x(i)<br /> <br /> l= −h+2 i=k+l−1<br /> <br /> i<br /> T11<br /> = Q − P + P1 + P1T ,<br /> <br /> k−1<br /> k<br /> X<br /> X<br /> <br /> V5 (x(k)) =<br /> <br /> i<br /> T12<br /> = ΩT − ATi Ω + P1T ,<br /> <br /> xT (i)Sx(i)<br /> <br /> j=k−h i=j<br /> <br /> i<br /> T13<br /> = −ΩT Bi + P1T − P1 + P2T ,<br /> <br /> +<br /> <br /> i<br /> T14<br /> = P2T − P1 + P1T ,<br /> <br /> k<br /> X<br /> <br /> k−1<br /> X<br /> <br /> xT (i)T x(i)<br /> <br /> j=k−h i=j<br /> <br /> i<br /> T22<br /> = P + ΩT + Ω,<br /> <br /> We have<br /> <br /> i<br /> T23<br /> = −ΩT Bi − P1 ,<br /> i<br /> T33<br /> = −H + λHiT Hij − P2 − P2T − P1 − P1T ,<br /> i<br /> T34<br /> = −P1 − P1T − P2 − P2T ,<br /> <br /> V (x(k)) ≥ λmin (P ).<br /> The difference of V1 (x(k)) gives<br /> ∆V1 (x(k)) = V1 (x(k + 1)) − V1 (x(k))<br /> <br /> i<br /> T44<br /> = −P2 − P2T − P1 − P1T ,<br /> <br /> = xT (k + 1)P x(k + 1) − xT (k)P x(k).<br /> <br /> The following result gives what conditions have<br /> to be satisfied to guarantee that the system<br /> (2.1) is stable.<br /> <br /> Let us denote x(k + 1) = y(k), and<br /> <br /> Theorem 3.1. System (2.1) is asymptotically stable if there exist symmetric positive definite matrices P, Q, R, S, T, H, W, and a matrices Ω, P1 such that the following LMIs hold:<br /> <br /> <br /> P<br /> Ω<br /> Γ=<br /> 0<br /> 0<br /> <br /> H ≤ W,<br /> i<br /> <br /> T < 0, i = 1, 2, ..., N,<br /> <br /> (3.4)<br /> <br /> zi (k) = [x(k), y(k), x(k − h(k)), fi (.)],<br /> 0<br /> Ω<br /> 0<br /> 0<br /> <br /> 0<br /> 0<br /> I<br /> 0<br /> <br /> i = 1, 2, ..., N ;<br /> <br /> 0<br /> 0<br />  , We have<br /> 0<br /> I<br /> <br /> (3.1)<br /> <br /> ∆V2 (x(k)) = xT (k)(Q + R)x(k)<br /> <br /> (3.2)<br /> <br /> − xT (k − h)Rx(k − h) − xT (k − h)Qx(k − h).<br /> (3.5)<br /> <br /> ≤(1 + h)xT (k)T x(k)<br /> <br /> The difference of ∆V3 (x(k)) gives<br /> ∆V3 (x(k)) = V3 (x(k + 1)) − V3 (x(k))<br /> <br /> −<br /> <br /> = [xT (k)Hx(k) − xT (k − h(k))Hx(k − h(k))<br /> <br /> j=k−h<br /> <br /> + (1 + h)x (k)Sx(k)<br /> <br /> X<br /> <br /> xT (i)Hx(i)<br /> −<br /> <br /> i=k+1−h(k+1)<br /> k−1<br /> X<br /> <br /> +<br /> <br /> j=k−h<br /> <br /> T<br /> <br /> k−h<br /> <br /> +<br /> <br /> k<br /> k<br /> X<br /> X<br /> 1<br /> (<br /> x(j))T T (<br /> x(j))<br /> 1+h<br /> <br /> k−1<br /> X<br /> <br /> xT (i)Hx(i) −<br /> <br /> i=k+1−h<br /> <br /> k<br /> k<br /> X<br /> X<br /> 1<br /> (<br /> x(j))T S(<br /> x(j)).<br /> 1+h<br /> j=k−h<br /> <br /> xT (i)Hx(i)].<br /> <br /> i=k+1−h(k)<br /> <br /> (3.6)<br /> <br /> Since 0 ≤ h ≤ h(k) ≤ h, ∀k ∈ Z+ and H ≤ W,<br /> we have:<br /> k−1<br /> X<br /> <br /> The difference of ∆V4 (x(k)) gives<br /> <br /> k−1<br /> X<br /> <br /> xT (i)Hx(i) ≤<br /> <br /> i=k+1−h<br /> <br /> ∆V4 (x(k)) =<br /> <br /> j=k−h<br /> <br /> i=k+1−h(k)<br /> <br /> k−h<br /> <br /> X<br /> <br /> T<br /> <br /> = (h − h)x (k)W x(k) −<br /> <br /> k−h<br /> <br /> X<br /> <br /> k−h<br /> T<br /> <br /> x (l)W x(l).<br /> <br /> l=k+1−h<br /> <br /> i=k+1−h(k+1)<br /> <br /> X<br /> <br /> X<br /> <br /> xT (i)Hx(i) ≤<br /> <br /> xT (i)Hx(i);<br /> <br /> i=k+1−h<br /> <br /> k−h<br /> <br /> (3.7)<br /> <br /> xT (i)Hx(i);<br /> <br /> k−h<br /> <br /> X<br /> <br /> xT (i)Hx(i) ≤<br /> <br /> i=k+1−h<br /> <br /> xT (i)W x(i).<br /> <br /> i=k+1−h<br /> <br /> Using Lemma 2.3, we have<br /> <br /> k+1<br /> X<br /> <br /> ∆V5 x(k) =<br /> <br /> k<br /> X<br /> <br /> (3.9)<br /> Let ν(k) = x(k + 1) − x(k), we obtain x(k) −<br /> k−1<br /> P<br /> [<br /> ν(i) + x(k − hj (k))] = 0, thus, for<br /> <br /> T<br /> <br /> x (i)T x(i)<br /> <br /> i=k−hj (k)<br /> <br /> j=k+1−h i=j<br /> <br /> −<br /> <br /> k<br /> k−1<br /> X<br /> X<br /> <br /> arbitrary matrices P1 , P2 with appropriate dimensions, we have<br /> <br /> <br /> 0 P1<br /> T<br /> X<br /> Y = 0,<br /> (3.10)<br /> 0 P2<br /> <br /> xT (i)T x(i)<br /> <br /> j=k−h i=j<br /> k+1<br /> X<br /> <br /> +<br /> <br /> k<br /> X<br /> <br /> xT (i)Sx(i)<br /> <br /> where<br /> <br /> j=k+1−h i=j<br /> <br /> −<br /> <br /> k<br /> k−1<br /> X<br /> X<br /> <br /> X T = (ξ T (k), [<br /> <br /> T<br /> <br /> x (i)Sx(i)<br /> <br /> k−1<br /> P<br /> <br /> ν T (i) + xT (k − h(k))]),<br /> <br /> i=k−h(k)<br /> <br /> j=k−h i=j<br /> <br /> =<br /> <br /> k<br /> X<br /> <br /> (xT (k)T x(k) − xT (j)T x(j))<br /> <br /> +<br /> <br /> (xT (k)Sx(k) − xT (j)Sx(j))<br /> <br /> j=k−h<br /> <br /> =(1 + h)xT (k)T x(k)<br /> −<br /> <br /> k<br /> X<br /> <br /> xT (j)T x(j)<br /> <br /> j=k−h<br /> <br /> + (1 + h)xT (k)Sx(k)<br /> −<br /> <br /> k−1<br /> P<br /> <br /> ν T (i) + xT (k −<br /> <br /> i=k−h(k)<br /> <br /> h(k))]),<br /> <br /> j=k−h<br /> k<br /> X<br /> <br /> (3.8)<br /> <br /> Y T = (y T (k), xT (k) − [<br /> <br /> k<br /> X<br /> j=k−h<br /> <br /> xT (j)Sx(j)<br /> <br /> ξ(k) = (x(k) + y(k) + x(k − h(k)) + x(k − h) +<br /> k<br /> k<br /> k−1<br /> P<br /> P<br /> P<br /> x(k−h)+<br /> x(i)+<br /> x(i)+<br /> ν(i)+<br /> i=k−h<br /> <br /> i=h+h<br /> <br /> i=k−h(k)<br /> <br /> f (.)).<br /> We note that condition (2.2) equivalent to<br /> <br /> <br /> −GTi Gi 0<br /> 0<br /> 0<br />  0<br /> 0<br /> 0<br /> 0<br />  z (k) ≤ 0<br /> ziT (k) <br /> T<br />  0<br /> 0 −Hi Hi 0 i<br /> 0<br /> 0<br /> 0<br /> I<br /> (3.11)<br /> <br /> − xT (k − h)Rx(k − h) − f T (.)λIf T (.)<br /> <br /> From (3.5)-(3.12) it follows that<br /> ∆V (xk ) ≤ xT (k)Li (R, Ω, H, W, T, S)x(k)<br /> +<br /> +<br /> <br /> −2<br /> <br /> T<br /> <br /> i T<br /> i<br /> x (k)(k)T11<br /> x (k) + 2xT (k)T12<br /> y(k)<br /> T<br /> i<br /> 2x (k)T13 x(k − h(k))<br /> <br /> + 2x<br /> <br /> T<br /> <br /> i<br /> (k)T14<br /> <br /> k−1<br /> X<br /> <br /> k<br /> X<br /> <br /> +<br /> <br /> ν(i)<br /> <br /> k<br /> X<br /> <br /> x(i) +<br /> <br /> i=k−h<br /> <br /> x(i)),<br /> <br /> i=k−h<br /> <br /> where<br /> k<br /> X<br /> <br /> + 2xT (k)P1T x(k − h) + 2xT (k)P1T<br /> <br /> x(i)<br /> <br /> i=k−h<br /> <br /> + 2xT (k)P1T<br /> <br /> ν(i)P1T (x(k − h)<br /> <br /> i=k−h(k)<br /> <br /> i=k−h(k)<br /> <br /> k<br /> X<br /> <br /> k−1<br /> X<br /> <br /> x(i) + 2xT (k)P1T x(k − h)<br /> <br /> i=k−h<br /> <br /> λ > 0,<br /> i<br /> i<br /> T11<br /> = Q − P + P1 + P1T , T12<br /> = ΩT − ATi Ω + P1T ,<br /> ij<br /> T13<br /> = −ΩT Bij + P1T − P1 + P2T ,<br /> i<br /> T14<br /> = P2T − P1 + P1T ,<br /> <br /> i<br /> + 2xT (k)(−Ω + P1T )f (.) + 2y T (k)T22<br /> y(k)<br /> <br /> ij<br /> i<br /> T22<br /> = −ΩT Bij − P1 ,<br /> = P + ΩT + Ω, T23<br /> <br /> + 2y T (k)(−ΩT Bi − P1 )x(k − h(k))<br /> <br /> ij<br /> T<br /> = −H + λHij<br /> Hij − P2 − P2T − P1 − P1T ,<br /> T33<br /> <br /> − 2y T (k)P1<br /> <br /> k−1<br /> X<br /> <br /> ν(i) − 2y T (k)ΩT f (.)<br /> <br /> i=k−h(k)<br /> ij<br /> + 2xT (k − h(k))T33<br /> x(k − h(k))<br /> <br /> k−1<br /> X<br /> <br /> i<br /> T44<br /> = −P2 − P2T − P1 − P1T ,<br /> <br /> We can verify that<br /> <br /> + 2x(k − h(k))(−P1 )x(k − h(k))<br /> i<br /> + 2xT (k − h(k))T34<br /> <br /> i<br /> = −P1 − P1T − P2 − P2T ,<br /> T34<br /> <br /> ∆V (xk ) ≤ xT (k)Li (R, Ω, H, W, T, S)x(k)<br /> <br /> ν(i)<br /> <br /> + ϕi (k)T i ϕi (k)<br /> <br /> i=k−h(k)<br /> <br /> (3.13)<br /> <br /> − 2xT (k − h(k))P1T x(k − h)<br /> k<br /> X<br /> <br /> − 2x(k − h(k))T P1T<br /> <br /> where<br /> <br /> x(i)<br /> <br /> h1 (k)),<br /> <br /> i=k−h<br /> k<br /> X<br /> <br /> − 2xT (k − h(k))P1T<br /> <br /> ϕ(k)<br /> k−1<br /> P<br /> <br /> =<br /> <br /> x(i)<br /> <br /> ν(i),<br /> <br /> i=k−h(k)<br /> k−1<br /> X<br /> <br /> −2<br /> <br /> ν(i)<br /> <br /> i=k−h(k)<br /> <br /> i = 1, 2, ..., N.<br /> We now apply the condition (3.4) and Lemma<br /> (2.2), the system Li (R, Ω, H, W, T, S) is strictly<br /> complete and the sets Ωi and Ωi are well defined<br /> such that<br /> <br /> ν T (i)P1T f (.)<br /> <br /> i=k−h(k)<br /> T<br /> <br /> N<br /> [<br /> <br /> − x (k − h)Qx(k − h)<br /> −<br /> <br /> k<br /> X<br /> i=k−h<br /> <br /> −<br /> <br /> k<br /> X<br /> i=k−h<br /> <br /> 1<br /> x (i)<br /> T<br /> 1+h<br /> T<br /> <br /> k<br /> X<br /> <br /> x(i), f (.))<br /> <br /> i=h+h<br /> <br /> ∆V (xk ) < xT (k)Li (R, Ω, H, W, T, S)x(k),<br /> <br /> k−1<br /> X<br /> <br /> ν T (i)P1 x(k − h)<br /> <br /> k−1<br /> X<br /> <br /> k<br /> P<br /> <br /> Therefore, we finally obtain from (3.15) and<br /> the condition (3.3) that<br /> <br /> i=k−h(k)<br /> <br /> −2<br /> <br /> x(i),<br /> <br /> i=k−h<br /> <br /> − 2xT (k − h(k))P1T f (.)<br /> i<br /> ν T (i)T44<br /> <br /> k<br /> P<br /> <br /> x(k − h), x(k − h),<br /> <br /> − 2x(k − h(k))T P1T x(k − h)<br /> k−1<br /> X<br /> <br /> −<br /> <br /> i=k−h(k)<br /> <br /> i=k−h<br /> <br /> +<br /> <br /> (x(k), y(k), x(k<br /> <br /> i=1<br /> <br /> x(i)<br /> <br /> i=k−h<br /> <br /> Ωi = Rn \{0},<br /> <br /> N<br /> [<br /> <br /> Ωi = Rn \{0}, Ωi<br /> <br /> i=1<br /> <br /> i 6= j.<br /> <br /> k<br /> X<br /> 1<br /> T<br /> x (i)<br /> S<br /> x(i)<br /> 1+h<br /> i=k−h<br /> <br /> (3.12)<br /> <br /> \<br /> <br /> Ωj = ø,<br /> <br />